Functional renormalization group for extremely correlated electrons

TL;DR

This work develops an X-operator functional renormalization group (X-FRG) to study the Hubbard model in the strict $U=\infty$ limit, focusing on the $t$ model with projected Hilbert space and non-canonical holon algebra. By deforming hopping with $t_{\Lambda,ij}=\Lambda t_{ij}$ and treating local physics exactly, the authors obtain non-perturbative, thermodynamic-limit insights into extreme correlations, including a phase diagram featuring Nagaoka ferromagnetism at high densities, stripe antiferromagnetism at intermediate densities, and a paramagnetic Fermi liquid at low densities, along with a bad-metal spectral character in magnetically ordered regimes. The study reveals substantial renormalization of the bandwidth, strong particle-hole asymmetry, and pronounced polaronic continua in the hole sector, accompanied by systematic violations of Luttinger's theorem in certain densities. The X-FRG framework thus provides a powerful, unbiased tool for exploring extreme correlation physics directly in the thermodynamic limit and suggests promising extensions to the full $t$-$J$ model and bosonic channels, with Ward identities and spectral normalization as important future directions.

Abstract

At strong on-site repulsion $ U $, the fermionic Hubbard model realizes an extremely correlated electron system. In this regime, it is natural to derive the low-energy physics with the help of non-canonical operators acting on a projected Hilbert space without double occupancies. Using a strong-coupling functional renormalization group technique, we study the physics of such extreme correlations in the strict $ U = \infty $ limit, where only kinematic interactions due to the Hilbert space projection remain. For nearest-neighbor hopping on a square lattice, we find that the electronic spectrum is significantly renormalized, with bandwidth and quasi-particle residue strongly decreasing with increasing electron density. On the other hand, damping and particle-hole asymmetry increase, while a polaronic continuum forms in the hole sector, below the single-particle band. Fermi liquid phenomenology applies only at low densities, where the system remains paramagnetic. At higher densities, we find a bad metal with strong magnetic correlations, indicating that the ground state is the Nagaoka ferromagnet at high densities and a stripe antiferromagnet at intermediate densities. Both in the paramagnetic and the ferromagnetic regimes, we observe a violation of Luttinger's theorem.

Figures (25)

• Figure 1: Temperature-density phase diagram of the $t$ model obtained from the numerical solution of the X-FRG flow equations. Colored symbols show the temperature where the flow breaks down for a given density $n$. Black circles indicate the temperature below which the system becomes conducting. Connecting lines are guides to the eye. At densities $n < n_{ c , 1 } = 0.48$, the $t$ model remains a paramagnetic (PM) Fermi liquid (FL). Flow breakdown in this regime, as well as for $n \gtrsim 0.95$ (gray circle), is due to the finite number of Matsubara frequencies in our numerical implementation. For $n \ge n_{ c , 1 }$ the $t$ model exhibits non-Fermi liquid behavior (nFL). The flow breaks down in this density range because of physical instabilities indicative of magnetically ordered ground states---a stripe antiferromagnet for $n_{ c, 1 } \le n < n_{ c , 2 } = 0.6$, and two disinct ferromagnets for $n_{ c , 2 } \le n < n_{ c , 3 } = 0.85$ (FM1) and $n_{ c , 3 } \le n < 1$ (FM2). The latter are separated by a Lifshitz transition of the electronic Fermi surface.
• Figure 2: Diagrammatic flow equations of (a) holon self-energy $\Sigma_\Lambda ( K )$ and (b) two-body interaction vertex $U_\Lambda ( K_1' , K_2' ; K_2 , K_1 )$. The holon propagator $G_\Lambda ( K )$ is represented by a line with an arrow. If the line is slashed, it has to be replaced by the corresponding single-scale propagator $\dot{G}_\Lambda ( K ) = \cos ( \omega 0^+) G_\Lambda^2 ( K ) \partial_\Lambda ( t_{ \Lambda , \bm{k} } - 2 \delta \mu_\Lambda / 3 )$. Closed loops with a cross inside mean a sum of diagrams where each $G_\Lambda ( K )$ is in turn replaced by $\dot{G}_\Lambda ( K )$.
• Figure 3: Diagrammatic flow equations of (a) the superconducting channel $\mathcal{S}_\Lambda(Q_\textrm{pp}; K_2',K_1)$, (b) the magnetic channel $\mathcal{M}_\Lambda(Q_\textrm{ex}; K_2,K_1)$, and (c) the charge channel $\mathcal{C}_\Lambda(Q_\textrm{fs}; K_2,K_1)$.
• Figure 4: Momentum dependence of the static interaction channels within the first Brillouin zone, evaluated for four selected densities and corresponding lowest temperature accessible via the truncated flow equations. First row: magnetic channel $\mathcal{M}_\Lambda(\bm{q};0,0)$; second row: charge channel $\mathcal{C}_\Lambda(\bm{q};0,0)$; third row: superconducting channel $\mathcal{S}_\Lambda(\bm{q};0,0)$.
• Figure 5: (a)-(c) Static magnetic susceptibility $\chi ( \bm{q} , 0 )$ at momenta $\bm{q} = \bm{q}_{ \textrm{FM} } = ( 0 , 0)$ (solid lines) and $\bm{q} = \bm{q}_{ \textrm{stripe} } = ( \pi , 0)$ (dashed lines) in the low-density paramagnetic, intermediate stripe, and large-density ferromagnetic regimes, respectively. (d) Diagrammatic representation of the dynamical spin susceptibility $\chi ( Q )$. (e)-(f) Effective exchange couplings for stripe and ferromagnet, extracted from the exponential rise of the associated susceptibility immediately before the flow breaks down. Lines are guides to the eye. The color scheme follows the phase diagram shown in Fig. \ref{['fig:PhaseDiag']}.